Help with math and thinking intuitively about it.

[RichardParker]

I'm studying math proofs and I can't seem to grasp the intuition behind various proofs.

Does this mean I don't understand the mathematics behind it, even though I understand perfectly the logic?

Is mathematics supposed to be intuitive?

Edit: I know basic facts of logic is founded on reason, and these are all intuitive. But when the it gets tough, the intuition gets going.

 

[micromass]

It depends on which proof it is. Some proofs are very technical, and don't seem to have much intuition. For example, proving that sin(2x)=2sin(x)cos(x) is quite easy, but I don't seem to have much intuition in the proof.

On the other hand, there are a lot of proofs which DO have some underlying idea. And once you grasp this idea, the proof becomes very easy. But sometimes recognizing this idea requires a lot of time and experience.

My favorite quote regarding mathematics is "In mathematics you don't understand things, you just get used to them" by Von Neumann. I think it is true to some extent. And if you practise a lot on proofs, then after a while it comes naturally.

Can you tell us which proofs you are having troubles with? Maybe we can tell you what the intuition is...

 

[mathwonk]

many books re written without giving any intuition, just the technicalities. i hate that. try other books. e.g. rudin's books, give in my opinion no intuition at all.

 

[A. Neumaier]

Is mathematics supposed to be intuitive?

No. It is supposed to be rigorous.

Intuition is a useful and important extra mode to assess the quality/value/usefulness of mathematics, not its correctness. It helps you to understand how to apply mathematics, how to choose between different ways of doing something, how to remember the main facts without having to remember all details, how to organize a subject, whether you should look at a particular piece of research or at the details of a proof, etc.. Typically, the more (good) intuition you have about a topic the less complex the matter appears, and the easier it is to quickly see what goes on.

But different mathematicians may have very different intuitions for the same subject matter; so this is a personal and somewhat subjective thing. That's why some writers (e.g., Bourbaki) minimize the amount of intuitive guidance - to be readable, such material must be very well organized then.

 

[mathwonk]

maybe that's why i always felt like a fish out of water. my friends told me i should have been in social studies.

 

[allispaul]

Is mathematics supposed to be intuitive?

I'd say yes, but not in the way you'd think. Intuition is a mathematician's most valuable asset -- it helps you make connections, organize your knowledge, and develop it into new knowledge. When you're given an unfamiliar statement and asked to prove it, it's your intuition that's doing the real work. In early proof-based classes, this will often consist in recognizing the cues that the question is giving you and its relation to what you've studied in class, but that's okay. It's training for later, when you might be trying to prove statements that nobody else has even considered, and your intuition is the tool that will blaze possible trails for you to follow back to your previous knowledge.

Of course, proofs must be rigorous, but a computer can manage rigor. What you should really be concerned with is whether your intuition meshes with logic -- and if not, why not. Mathematicians defined concepts (like groups) to formalize their intuition about fundamental ideas (like transformations). Thus, if you feel that something should be true when logic shows that it's false, come up with some examples (some weird examples, preferably) and figure out where you went wrong. Did you make a miscalculation? Were you thinking about the definition in the wrong way? Or was the definition itself inadequate? If you feel that it was inadequate, it's worth taking your time to try to construct a better definition and seeing if you meet contradictions or fail in some other way. In beginning math, the definition you're given is probably the right one, but exercises like this were the seeds out of which things like topology and non-Euclidean geometry grew.

On the other hand, it's entirely possible that specific proofs can be very non-intuitive. We proved the Unit Theorem in my number theory class today, and one major part of the proof was nothing but a string of calculations establishing bounds on a certain number. In cases like these, it's worth splitting the proof apart into pieces (say, every time a claim is made in order to be used later), and asking yourself at each step what the author was trying to do, why he/she wanted to do that, and how he/she did it. In my case, the aim was to bound that number, the reason was because it made a certain set finite, and the method was a geometric one, finding other numbers that it had to be far apart from.

If you can understand it at this level, then you're set. There's no need to understand the intuition behind each specific calculation. If you can't, then take a step back, identify the major forces in the proof, and try to make those intuitive. Even just look at the statement of the proof, make that intuitive, and figure out how it transforms inside the proof.

 

[mathwonk]

here i a quote from david hilbert's " anschaulich geometrie" translated as geometry and the imagination, but more accurately as perhaps "descriptive geometry" or "visual geometry".

"In mathematics as in any scientific research, we find two tendencies present. On the one hand the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them so to speak, which stresses the concrete meaning of their relations.

...It is till true today as it ever was that intuitive understanding plays a major role in geometry. And such concrete intuition is of great value not only for the research worker, but also for anyone who wishes to study and appreciate the results of research in geometry.

In this book it is our purpose to give a presentation of geometry as it stands today, in its visual intuitive aspects..."

 

[A. Neumaier]

here i a quote from david hilbert's " anschaulich geometrie"
''In this book it is our purpose to give a presentation of geometry as it stands today, in its visual intuitive aspects..."

Yes, this is a laudable purpose. I never liked the Bourbaki-style presentations, though I learned to understand them, and his tome on Lie algebras is excellent. In my own work, i strive for combining rigor and intuition...

 

[Johannes]

In my experience it also helps a lot to just try to really DO the proofs, when studying with a book. Read and "visualize" the statement and then think about how you would proof it. This helps (at least me) a lot in understanding the proof, because frames the problem much more and makes the steps in the proof less arbitrary, since I might have used different approaches which did not lead anywhere.